Inferring Cumulative Default Probability from Bond Yields and Recovery

Summary

The note outlines a simplified way to infer a risky issuer’s cumulative default probability from the yields on a risk-free zero-coupon bond and a risky zero-coupon bond with the same maturity. It assumes a fixed recovery rate and equates the discounted expected payoff of the risky bond, weighted between repayment and recovery, with its market value implied by the risky yield. Solving that relationship for the default probability gives an implied estimate.

The example concerns a ten-year Treasury strip and a corporate zero, with semiannual compounding and a stated recovery assumption, but the excerpt ends before reporting the numerical estimate. The setup relies on restrictive assumptions: a single cumulative probability, fixed recovery, and a simplified payoff at maturity. It does not model the timing of default, recovery uncertainty, liquidity differences, taxes, or risk premia embedded in credit spreads. The result should therefore be read as a model-implied quantity, not a direct forecast or observed default frequency.

Key ideas

• A risky zero-coupon bond’s yield can be compared with a risk-free yield to form an implied default estimate.
• The method represents the terminal payoff as full repayment without default and recovery following default.
• Equating expected discounted payoff with the bond value produces an equation that can be solved for cumulative default probability.
• The calculation assumes fixed recovery and simplifies default timing and credit-risk compensation.
• The example provides inputs but does not include the final numerical estimate.

Tags

# Calculating the cumulative probability of default from recovery rate, yield and coupon rate


# Calculating the cumulative probability of default from recovery rate, yield and coupon rate












I have the following details: A 10-year U.S.Treasury strip has a yield of 6% and a 10-year zero issued by XYZ Inc, rated A by S&P and Moody's, has 7% (semi-annual compounding). Assuming a recovery rate of 45% What is the cumulative probability of XYZ Inc, defaulting during next 10 years?

How do I calculate the cumulative probability?

## Answer by alexbougias (score 1, accepted)

https://quant.stackexchange.com/a/59934

Let us denote with $r_f$ the yield of the 10 U.S.Treasury strip and $r_{A}$ the yield of the risky bond issued by XYZ Inc. We denote with $p$ the cumulative default probability, with $P$ the bond face value, with $R$ the recovery rate and with $T$ the bond maturity. In the absence of arbitrage, we have

$$ \dfrac{(1-p)\times FV+p \times R \times FV}{ \left(1+\frac{r_f}{2} \right)^{2 \times T}}=\dfrac{FV}{ \left(1+\frac{r_A}{2} \right)^{2 \times T}}$$

Solving the above equation for $p$, we get the implied default probability. In your specific example, we get:

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.